Cube moves for $s$-embeddings and $\alpha$-realizations

TL;DR

This paper introduces the cube move transformation for $s$-embeddings, proves its existence and uniqueness, and generalizes the concept to $ alpha$-embeddings, providing new insights into planar graph tilings and the Ising model.

Contribution

It establishes the existence and uniqueness of the cube move for $s$-embeddings and generalizes the framework to $ alpha$-embeddings, unifying different embedding classes.

Findings

Proved the cube move is well-defined and unique for $s$-embeddings.

Derived a simplified formula for the Ising star-triangle transformation.

Extended the concept of embeddings to a broader class called $ alpha$-embeddings.

Abstract

Chelkak introduced $s$-embeddings as tilings by tangential quads which provide the right setting to study the Ising model with arbitrary coupling constants on arbitrary planar graphs. We prove the existence and uniqueness of a local transformation for $s$-embeddings called the cube move, which consists in flipping three quadrilaterals in such a way that the resulting tiling is also in the class of $s$-embeddings. In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation which is conjugated to the cube move for $s$-embeddings. We introduce more generally the class of $\alpha$-embeddings as tilings of a portion of the plane by quadrilaterals such that the side lengths of each quadrilateral $ABCD$ satisfy the relation $AB^\alpha+CD^\alpha=AD^\alpha+BC^\alpha$, providing a common generalization for harmonic embeddings adapted to the study of resistor networks ($\alpha=2$) and for $s$-embeddings ($\alpha=1$). We investigate existence and uniqueness properties of the cube move for these $\alpha$-embeddings.